 today with Paulus for a lecture which is titled A Boundary Perspective on Physical Processes. Okay, thanks Yvannes. So the aim of the lectures today is to try to show how to compute physical processes in expanding universe, generally for the cosmologists, without having to go through the time integrals through which in principle they are the Feynman diagrams in perturbation theory are defined. And this would be very important also to understand a little bit, it's the structure of these processes. But before doing that, I think it's useful to make a small recap of the previous lecture. So what were the most important messages from the lecture of yesterday? The first one is that, so the bunch Davis wave function has singularities just in sums of energies. I'm putting energies in quotation mark because I think it has been clear it's a little bit of an abuse of language because there is not just such a thing as energy these are associated to modularity of the moment of the spatial momentum. And so the sums can be either can can involve either all the the energies of the process or some of the external energies and the some of the internal energies, which implies that second important message they correspond to energies to the total energy of sub processes. For example, what it means what we saw is that if for example I take the process in which from the vacuum we get a four-pointed state with an exchange, this line as usual is the boundary at the infinite future. And these are our external states, this is the internal one, the type of singularity that we can have involve either the sum of the external energy, so the total energy of the process, so involve the full process, so this will be like u1 plus u2 plus u3 plus u4 equals zero, then we can have another sub process might involve just these vertex, so we'll have u1 plus u2 plus y equals zero, well let me call that y is the final p1 plus p2, which is by spatial momentum conservation is equal to p3 plus p4. And similarly, the sub process is associated to these other vertex where then the condition will be that y plus p3 plus c4 is equal to zero. So this process will have singularities at these points. And another important piece of information is that all the singularities lie outside the physical region. Well, the physical region was defined by having all the energy positive as well as all the internal energies positive, but this stays for internal, because in this case we have just one, but for more complicated processes we have many, but all of them are in principle city positive. So if we stay in the physical region, you'll know whether you can satisfy these conditions that they are individually zero, but then you get trivial processes, so you don't get anything interesting, you get zero. So you don't have any process. So in order to access them, you need to do a summary continuation to decide that some of the energies can become negative. And in the case, what one is doing is actually rather than having just one type of state, one we have both in and out state. And for that reason, another thing that I showed in a concrete example, but it's generally true, as actually you could hear when Nima answered my question at the very end of his talk, that it's a general fact that then the coefficients of the singularities are associated to the high energy limit of flat space scattering. So in this case, for example, what will happen is that when you get close to this point, you not only have in an upstate if you do this energy continuation in order not to get something trivial, but then this condition means also that you are restoring energy conservation. And therefore, what you have is that your wave function for this process will be something like the contribution to the ampers associated with the same graph. This is a G which stays for the particular graph. And then here you have the singularity, which depends on the specific background and states that you have. Sure. Okay, very good. Because the way that this singularity comes about is because when you have this time integrals from minus NP to zero of eta times eta, you have this type of structure. What's happening is that you observe the singularity corresponds to when in these integrals you go close to the region where eta goes to the infinite past. And so what's happening is that when eta goes in this region, if you don't do anything, you will get this which goes to zero because you have to you have to introduce some IEPs on to regulate it. However, if your energy is zero at the point in which this energy is zero, rather than having this exponential damping, you will have an actual singularity. So the point is that the high energy comes from the fact that you are moving. So you're giving that you have this in this region. So having the total energy equals zero means that you are taking all the process arbitrary for when the past. So you are taking it in the high energy region because in the infinite past you go in at a higher high energy. That's motivation. This also means that you have some massive state. You're going to get an amplitude for the corresponding massless state. And actually, this is actually you can make this type of argument even for when you rather than considering the total energy, you consider just the total energy of a sub process. And then you can see this, the fact that when, for example, you're here, at this point, this again, you have a factorization between the amplitude associated to the sub process. Let me call it small g times. And then you have, let me call it now, psi tilde, which is a wave function associated to this other sub process. So actually, let me call it psi tilde of g bar. So which is the complement graph times the singularity, which and then you have singularity of some form. Okay. So why one was getting this factorization, one was getting factorization because in this limit, if you remember the propagator, this bulk to bulk propagator as three terms, one which correspond to the retarded part, the other one to the advanced part, and the other one to the boundary condition. And so what's happening is that in this limit, just one of these two terms will contribute. So let's say that contributes the retarded part. And therefore, what one gets is that here, this contains the mode function associated to this leg with negative energy in this case, with positive energies in this case. And so what one gets is that this psi tilde that I wrote here is really one over twice the real part of the two, which is a factor which is in front of each of this term. And then here one get the three level, the small process with the minus the energy, we come from here minus, which is this minus. And then the same process, but now the internal energy which goes to the boundary, which is now is described by the mode function containing each of these pieces as now here positive energy. Okay, so one discovered that there are some sort of factorizations. Okay, so like in the Scatrina amplitude where we know that when we cross singularities, we get the amplitude factorizing smaller in a product of smaller lower point amplitudes. In this case, you see that when also you cross this singularity of sub processes, your wave function factorizes, but now factorizes in a part which is associated to the flat space process. And a part which is still, let's say, cosmological because it's given by the wave function. Okay, so this is very important because this gives you an idea of what is the physical content, which is encoded in the coefficients of the singularities, which in a sense, for example, if let's assume that we know very well what's the singularity structure of our wave function, we can actually use this information to try to bootstrap the wave function without having to do this time integrations. Okay, very good. There is any question about this? Okay, it's not so far. I mean, what I say here are general properties with the only caveat that I'm making the assumption that the states I'm considering I have a flat space counterpart, which, for example, if you are caring about the sifter, it means that there is a regular contraction for your sifter representation to some Poincaré representation, the unit representation. That's the only underlying assumption. Yeah, no. Well, this, okay, let's say that you have some state which fits in this condition, I told you, but also you have some interactions of the type that you said. So if you have just this type of non-local interaction, then obviously the structure will be different because you have some non-local singularities. And therefore, you might or might not have this pole. So in a sense, you can, sorry, this is still valid, but just the answer is that when you try to go to this point, you get zero because you don't have this flat space counterpartous process. Well, if you understand what's the meaning of this singularities that this non-local interaction introduce, then you might. For the moment, I don't have... No, that's correct, but that just means that when you go to this point that I was talking about, either you get zero or you get a softer singularity with a coefficient that can be some other process, something which is a purely cosmological effect. And then you have extra singularities which are given by the non-local term. And if you're asking the question, can I bootstrap it, it's really important that you understand what's the meaning of this extra singularity that the non-local is introducing and what's happened when you approach it. But in that case, you need to be careful because sometimes I think... I'm not sure if it was the case of Imaldus and Uspaper, but sometimes you can get these terms, but then you can make a field definition and you can get rid of them. If you can do that, this means that you don't really have this local term. I think if you set this example, the opportunity to get on a taking path, to be able to take the exchange, so the exchange here, like there's an outside advice that you need to get by taking the softer one. Yes. Four point function has a well defined attitude in it. And when you take the softer, then you just exchange the non-dynamic function. There has some attitude in certification when you exchange the exchange. Yes, it's not an attitude in the usual sense, but there's so often an interpretation when you exchange the sort of non-dynamic part of the engine. Yeah, I think if you are asking the question about can I bootstrap this thing, it's important what we said that you really need to have an interpretation or what this coefficient would be or what happened when you... So either in the total energy pool or what happened when you take these points. Okay, so, yeah, so what... Yeah, I was saying, so far this, what I was saying is general up to some caveat through which I have tried to be explicit about it. Now, let's to understand a little bit more how we can do some computation. Unfortunately, we cannot be a super general, so I think I will restrict to a class of scalar model, which are described by the following action. So it looks like a massless states in flat space, but with time-dependent couplings. So this type of model actually, if I declare that I have my this time-dependent coupling is given by some number, which is the coupling constant, times 2 plus 8 minus 1 divided by 2, where this an is the work factor in my FRW cosmology in conformal time, then this model is describing a conformal couple scalar with the in arbitrary FRW cosmologies and with polymer interaction. Now, you can also generalize it by inserting a term, which is like a mass term, which depends also on time. Okay, and again, upon identification of this time-dependent mass with some function of the work factor, this actually will describe a general massive state in FRW cosmology. Okay, I will restrict for the moment to this case for two main reasons. The first one is that there is for some specific massive states, we can actually recover the answer for the function of the conformal couple. So I can compute the conformal couple, then do some operation and obtain that information. The second reason is that for general masses, which means if you are in the center for states which are in the principle series of the unitary reduced representation of the center, which correspond to very heavy masses, then there is no real simplification, the calculation, you get some general some hanker function whose order parameter is purely imaginary, and then there is, as I said, there is no simplification in the computation. So even that I won't try to argue how to, so I want to, so I will just restrict myself to those cases where it's clear how to simplify the computation and therefore how to make actually this computation direct from the boundary. Okay, we're here from the boundary, I just mean that what you want to know are just the information about, so typically this is our physical process, your boundary information are the moment, okay, and then maybe the mass of the state which propagates so the spectrum of your theory. Okay, so these are the only the information that you want to have in order to do the computation, and the computation from the boundary as I said one more time means not to have to do the timing integrals or which the Feynman diagrams are defined. Okay, so for example this Feynman diagram for this for this model will be given by an integral from minus infinity to zero, let me call this point A and B, the eta, eta, eta B, then I have now a non-trivial vertex function, so I need to put here some lambda, in this case I'm taking a cube, so lambda 3 eta, A lambda 3 eta B. Then now given that the free part here is like a flat space master states, my external states are described by just plain wave, so my mod function which correspond to the back-to-boundary propagator are just exponential, so I get e to the i e1 plus e2 eta A eta i e3 plus e4 eta B, and then I have the back-to-boundary propagator here, g of y eta A eta B, which let me remind its explicit form was 1, so y e minus i eta A minus eta B eta A eta B plus, okay, so in principle, now obviously I should specify what are these functions, so I should tell you which cosmology I'm taking in order to actually do this computation. However, there is one of the goals also is to try to understand which properties of a process might actually be common to a larger class of cosmologies, so in order to avoid to specify the cosmology until the very end of the computation, one thing that one can do is to take these couplings that are time dependent, I'm not specifying what is this function, but I say that I want to consider an integral, okay, let me write it a little bit slower, I want to consider an integral representation, I'm just doing that, if I do this, if I do this, then this integral becomes, and now I have my time integrals, let me for simplicity, you see that here the energy is associated to the states at which go to the same vertex, enter always in this combination, so I will call this big x1, this other one big x2, so now I get here e to the i big x1, but now because of this integral representation, I have also this exponential, so one for each time integration, so this x1 is just shifted by this zeta a, and the same for the other, and yes, just the one sec, and then the bark propagator, yes, here, yeah, because this goes back to the original condition that you want a bunch of this wave functions, so you have to pick the solution of your equation of motion, which are exponentially suppressed when you go to minus infinity, and which have a positive energy solution, and these are why you get this, so it depends on the choice, that's actually one of the reasons for which also, when you do the computation for the wave function, you also get a positive combination among the energies, if you were to have different signs there, then that would also mean that you have in an out state, because the sign of the energy is also telling you in which direction the state is going. Can you say that again? Yes, okay. Okay, actually that's, are you actually saying that any polynomial interaction is actually non-normalizable? I don't know, sure, no, I mean, okay, let me put in here, actually the segment also depends on the dimension, here I'm trying to make a general picture in which I'm considering arbitrary dimension and anything that in principle you can add that, then obviously you have to be careful about the normalization issue, so you can maybe fix your computation to cases where your theory is normalizable, let's imagine that we are in one plus three, so in four dimension, if I choose this to be four, that's fine, okay, so I'm trying to keep myself as general as possible because there is some part of the computation that does not depend on this issue, but obviously at some point you need to make some choices and you need to be to specify what you're interested in and then all these issues that you are specifying coming, so yeah, that's fine, yeah, just the same, okay, um, yes, no, no, but I'm not, no, that's correct because in principle, you can do it, but if you want to go to introduce, to go from conformal example to massless, you have to introduce a time-dependent mass term, we will be specific for, no, no, but what I'm saying is that you need to have some specific form here that tells you that then also it becomes the massless state in the sitter, okay, so now you can do it, I do the only reason for which I'm restricting to this is that it's simpler and for example in the case you want to study the massless solution in the sitter, I can actually deduce them from the conformal couple by acting with some differential operator, okay, so that's the reason for which this is a simple enough model to make some point, then there is this generalization you can do to go to more general type of states of mass scalar states and about this, yes, for the type of interaction, you don't, so I canceled it, but the coupling in principle, but the independent coupling, the only thing that you require is that it has this form with this number here, okay, so this is a minus one, half, okay, so here this is the work factor for your, that's correct, the only thing is that in fact, I mean one of the whole point of doing this is that now you have this piece, this is probably what you want, you're trying to argue, that does not really depend on the specific cosmology and then the specificity of the cosmology will enter when you have to do this integration, so there is a number of information that are completely common to any type of cosmology and come from this, from this integral, yes. But then the NP-PAR company is not the main step in fact, if you think about the HR, as long as you're creating this approach in a state in that other world, you're good, you're very diverse company at this time. So even in fact, if it is not completely different, if you think about regularization Yeah, very good. Yeah, so. So, here. So, now, so the, the whole point now will be to try to develop techniques to actually compute this object. And then, at the beginning, do this, this computation choosing whatever cosmology you prefer in principle, this is probably this is where the expression that Guillermo was writing yesterday come from. So they come from. If you take the work factor. Well, this type of form, maybe some gamma. Okay. Where L is the length, the length scale of your of your space them. And when you do this free transform. What you get is something of this type of this of this form, which is from zero plus infinity is Z to some power beta minus one. Okay. So, now, but in general, you might decide that not to care about the specific the specific form and just trying to infer as much as possible from this integrant that actually if you realize it's really similar it's really like the flat space with function. Because again, all the specificity of the cosmologies in this in this captains here. And but in this case that become an integrant. Let me say a universal integrant for an FRW cosmology for an FRW with function. Okay. Now, another thing that one can do is then I can just make a shift of this variable of integration Z in such a way to absorb this X in in the measure. So let's say that let me declare that X one small X one is the day plus big X one and small X two is that big plus big X two. Then what you get is now this lambda tilde depends on small X one mouse, big X one. And then now we have something which really looks like explicitly what we saw for the in the example yesterday that it was an explicit example in the case of flat space with function. Very good. Now, the statement now is that you can. So another thing that is important to observe is that I've been drawing this type of graph. Okay. One, two, three, four. Now this small X one and small X two are still related to the sum of the energies at this point in this in the states that are the same vertex, which means that in any case this information come always in this in this type of information so I can actually avoid to have to make this drawing and just suppress the external the bulk to boundary line, map it just to this type of graph and attaching the label small X one small X two to the sites to the vertices and why which is the internal to the edge. Okay, and it's much it's it's gonna be actually easier actually to do with the when when you analyze this graph. And now the statement is that the way function can be computed by iteratively erasing an edge until you are as one of the time all of them. What does mean so how do you see this, let's see in I mean, I'm going to prove it in this case, but the proof is completely general but I don't want to get, I mean, I might be just confusing to do to the general case. So, I will I'm going to focus on this. Sorry, this is. I'm going to focus on this part. Now, let me take an operator, which is the total time translation. In this case, for this graph is just given by this to derivative. Now I can consider an integral. That is just the same integral I have in time. But now with this operator that is acting on the full integrand, which is equal to the small X one. Hey, maybe. Now, this are total derivatives and such that at the boundary, when you compute these things at the boundary G vanishes and when you compute it at minus infinity, the exponential vice so the actual result from this integral is zero. However, I can integrate by part, making this operator acting on the external state. So I have just X one plus X two and the very same integral, which is sizing, and then I have this guy at the operator acting on on the back to boundary or back to back propagator. That if you do the explicit computation with the, the explicit form you see that is going to just begin by minus. This is going to be equal to the I y eta a plus it a b. Okay, which was down to have minus. So this, these two integral now factorize. And one thing that you recognize is this is just is a bunch of exponential so it's, it's all external state. It's like a contact interaction. So what you're really having here is that this part is a travel graph so a point but now the energy is not just x a is x a plus y. And this is another contact interaction with the x b plus y, whose energy is x b plus y. So this equation tells you that x one plus x two. For this process, it just given by erasing the internal edge. And associating the energy to internet to both and its sense point, the end points of the edge. Okay, so in general, this is a much more general situation so I can take this other graph. Okay, which correspond just an exchange of two of two states and do the same computing by suggesting that x one that this is given by one over one plus two plus extreme. If you will have to reiterate this procedure. You will see that your 10 extra term expression, which is one plus why if I raise this this leg. Let me call this why one so why so three. Then you have this other sub process. Plus, and this also works a loop level for the integrand of the loop. So you're still they will have to do the look integration. Okay, so this actually can make can be made in terms of recursive expression for in terms of in a graph theory language in the sense that the way that you can for example compute this graph. By saying that you take a graph, and then you associate the total energy, you take a sub graph, you associate the total energy of that, or that sub graph, then you look at that sub graph. And then you take again, you divide it in a smaller smaller sub graph and you associate to them with a large what this means that in this case, the first graph is the first sub graph of the graph is the graph itself. And I set with the energy. Then I look at it. There is just a way in which I can divide this into sub graph, which is this too, and I multiply this by the energy of this sub graph this noise, why plus X one and X two plus why. Rather than having this three term expression that you will have had by doing the explicit computation through the time integrals, you have on the nose, the one term expression for this graph that you obtain when you massage it. And again, I can give you directly this this graphical rules and the is a recursion. So this is telling you substantially that you can compute recursively a graph. In turn, of some of products of a smaller graph. So if you know the information about the most basic building blocks, which will be the contact interactions, then going up once you break this in all the pieces then going backwards, you can actually reconstruct the full information of the way function associated to this graph. Okay. And this sense this is a way of computing from boundary perspective because the only thing that you have to know are this, this, this, this energies associated to the, to the graph that you want to compute. And then you just are dividing this is smaller is smaller subgraph, which again contain information associated to the external say the external informant the boundary information associated to those subgraphs. Okay, yes. Yes. No, this procedure give you the full result because what what is telling you is that let's say you said you want to compute the exchange. Okay. Now, the sentiment is that the exchange graph is equivalent to if you multiply it by the total energy. That is just equivalent to the product of the content to contact interactions, which are whose energies are given by the original energies shifted by the energy of the internal state. So what you're doing it's like erasing this, this leg and associated the energy in the internal energy to each of the of the contact term, because in the bigger terms is like you're saying, for example, if you're taking this this interaction is like you're saying, you are erasing this and being in this to the boundary, but with the energy with the same sign. Thank you. Thank you. So if I asked you what is the thing with four external life. I think I haven't understood what's the confusion. I think I haven't understood what's the confusion. Do you really have a contact? Yes. No, okay. Okay. Okay, let's say this procedure works in. Here in principle. Okay, obviously it depends on on what you want to do. If you're in Lagrangian, you think that you have or you want to know if a cube interaction plus ff for interaction, obviously this gives you the information about the the exchange part and then you will have another contribution which is the counter term associated to the to the but they the point is that they have different couplings. Okay, so they lambda three lambda four. So that you can also compete independently in the sense that you can do this computation and then add them together. No, no, no. No, no, no, no, no. Okay, no, wait a second. In BCFW, the fact that had happened with some emails in principle you have a 3.3 vertex and the point interaction for point interaction, but the coupling is the same. So the fact that factor is a gauge artifact that you can make a field definition, which you can absorb it. So there there is really one fundamental vertex, not not to here instead, if you want to say that here you have two different couplings, then you can I can always tell you okay for some lambda you do the computation for the other lambda you do the computation and then you have the relevant contributions. So it was this the problem David or Yeah, as I said, I'm taking the procedure that I did this taking the integral the graph for an arbitrary cosmology do this manipulation on the on on the on the time dependent couplings in such a way on or if you want to do the time dependent part due to the to the work factor in such a way that I can map it in an integral over the external energies of a flat space object. And now I'm playing with the flat space object which now works as the integrands of the of my final FRW process. Yeah, I mean, what you will do, you can put it here but that's doesn't really change. Yeah. Yeah, in this case. Just looking at this integrand. This actually is for me another motivation to split the computation in this way is divided precisely you're going to get always simple pulse. So all the analysis is way easier. But obviously, once you want to do. If you let's if you let's say you want to do the final computation in a given cosmology then you have to do those integration, and then the point becomes to see how that integration that you have to do. The control integration intersected the the eventual singularities that you can have there so you are in any case how the singularities of the integrand can get mapped in the singularities of the of the final transcendent function that you might get. Okay, so in a sense, if you do that, I think you will get some term, because if you see one of the feature of this is that. The degree. So you can get a rational function and the degree of the number of the polynomial numerator is always lower than the degree of the polynomial denominator which means that typically the information about the pulse is going to be enough to fix it if you do it for the definition, you're going to get some term, which actually can have not have the structure. And so, in a sense, this shows Canon, it's some can be some sort of a chemical choice of what's in the field of definition that you are considering. So you won't be able to see these other these other terms. In fact, for me. In conclusion, I have still this question in mind that for some cases that I recospecified in his talk. It's solved but for a general quantum field theory in cross space, at least I don't feel that it's solved of, you know, how can distinguish the different way function of different correlation function in the center or whatever for the cosmology which differ just by the definition, or what's the content of this, because I think the case that Eric you studied it's very specific case and so that statement it's true just for that case the fact that there is no really feel differential and ambiguous. Okay. Very good. Yes, sure. So, so far, this population has done is really nice. So now when you do the, you feel a couple of things to do right if you want to know the book thing. Yes. And they contain the information what is this morning. That's correct. You see, in general, to do them or it must give it. Okay, so when you do. Okay, let's divide the answer into part let's stick to the three level contribution. Okay. For the contribution, you have just this integration over this access to do. And obviously, the fickle thing with them depends on so the cosmology that you decide that you choose, let's choose for fixing ideas. A cosmology, which is who's defined by the work factor of this time. Okay, so we got my some positive number. Okay. That the measure of the this integral. So I can write actually this integral over x one to plus one over small x one, then become something of the type of small x one minus x one to some other powers by B minus two, and the same for the other. So, for one thing that we have shown that for some choices of this beta. Okay, there's always at least the equal one from it's it's equal or greater than one for some choices of the beta then actually this you can actually do this integral and from you get the poly logarithms, and there is actually a technology to extract those information, even without doing explicit this integrals, which is in terms of what's called symbols you can extract the symbols and then they are not defined uniquely but there is a consistency condition here to fix the ambiguity. Okay. Now, if you then give me some crazy function, there are some crazy measure, then, obviously, I don't know. But for reasonably enough type of cosmology. There is some systematics in which you can do this. Now, another second, the second part of questions that obviously this is a tree level at loops, you will have these integrations plus look integration. If you do this integration, you do the same you get just poly logarithms. But then you have to look integration and there is not really known yet I mean there is no much computation. But the point is that you are mapping a problem to another problem, we are for which we can think to actually recycle techniques that have been developed in the amplitude context. Okay. Anna, for our for if gamma is gamma is one bit in the center. If you have to also do any because if you are fight to the cube or something. So let's say that you have gamma is equal one. Then you have a fight to the cube interaction in a three plus one dimension. Then this is also one so that's that's easy and this actually where most of the test has been done, but then these other ones are also related to this. This is the easiest but then if you change this, you can have other value for this integer. This is still can be done. Obviously again, there are more or less difficulties depends on the value of this betas you might have the point is that for some values of this beta. You might get a yard divergence divergences are shared plus infinity, which actually corresponds even if for energies going to infinity, they are infrared divergences. There is some been that. This really is telling you. That comes from a Fourier transform of a of a coupling, but that X go to infinity really means that you are approaching the boundary are equal equal equal zero for you for the three son that actually send code in infrared divergence due to the large volume. Okay. The point is that in this case that has specified that is what we worked out mostly. You are safe because you don't have this problem. Okay, this integration or this plus infinity. You don't have these divergences. If you change that you have to deal with this infrared divergence. Okay. So, given that I think I have three minutes left. Think the here the moral of the story is that I will take for some large quite large class of scalar model. The way function in FFW cosmology map it in an integral over an integrand which is substantially a flat space integrand, if I flat space with function do manipulation on that integrand and then do the integration and the manipulation is really connected to a graph theory operate the graph operation on the graphs. Okay. Which are expressed in terms of a recursion relations. Okay. Which means that if you have a big graph, you have to erase an edge at the time and some of all the possibility that you can raise one edge. And this is very important because here what one thing that is making manifest with this is the fact that at each sub graph. There is a singularity associated with so this total energy singularity is associated to the sub graph which is the graph itself. And these other singularities that are associated to these other sub graphs. So there is a correspondence between which is very important sub graph and singularities of the way function. Okay, and then when you do the integration actually the singularities are going to be mapped to singularity integrated function but now rather than being the small axis and why we're going to be in the big axis which is actually the real external data. And I think I can stop here. If you have more questions ahead otherwise I guess so we can go for coffee. Sorry, sorry, sorry, sorry. Yes, but the cap. No, no, no, no, no. If you want to do integrals, then you have to specify the statement is that you have to do this integrals, you have to integrate a certain measure, this formula in general, but then if you want the concrete function, you have to say that con los cosmologías que tienes, que significa cap'n'stienes. Si, para que sea la lección de... Si. Que siempre elegí, cual lo regaba. Ah, si, dame... Ugula 1 sería de cita, por ejemplo. Dob... No, no, la lección de beta, yo creo que había entendido que... Ah, si, si. Si, si, si. No, y... No, no, correcto. O puesto beta por no escribirla el número grande, la expresión grande, porque si no, de nuevo, tú tienes este tipo de expresión. Entonces, si yo pongo esto, lo que tienes aquí de cap'n'stienes. No, es buena. Es gamma. Entonces, beta. Es la lección de... ¿Por qué quiere que el... No, es una de las cosmologías que interesa la gente. Es una posible elección. No tienes por qué hacerla, tú puedes... Si tú eliges algo así, por ejemplo, por alguna razón, tú quieres algo así, con gamma positivo. Realmente, los integrales lo puedes mapear inderibly. Entonces, es hasta más fácil. ¿Vale? Pero si, si tú me dices un arco tan intenso de inventarte cosas más complicadas, evidentemente, son integrales. Pero esto, decimos, es una cosmología razonable que la gente mira. No, es una elección. Es que si tú empiezas, es la lección. Sorry, let's switch English. If you take the actual action for conformity couple scalar in... with the polynomial interaction in the usual way, which means that you have your Lagrange. It's a realization of the fact that you have something like this in principle. And then here it's not, that doesn't depend on time. So here you have the conformal calving and then, for example, you have this. Now, one thing that you want to do, I want to put this in the formula that I wrote before. So in principle here, you have powers of A. So what you do is you have to put this in the formula that I wrote before. So in principle here, you have powers of A. So what you do is some redefinition like this in such a way that you absorb the depends on here, but then you transfer this gets cancelled. You transfer here here. So this number here is due to this mapping. So that particular choice of that number. Okay. So this original information that was just the constant. Yeah, it gets a function because what you do. Making it up. No, no, no, no, no, no, no, no, no, no, no. You can do it genetically in the sense that if you are not worried about cosmology, I want to study some problem with a time dependent coupling. And so you can do this in any case. That's one of the trick. Yes. Yes. Just one question about this differential operator that you do this. Is it is it really teach. I mean, is this the consequence of the fact that you could, you know, like take the graph and the boundaries of action and modify anywhere on the time chain. No, it's it's really a consequence of the fact that you are using this is a time translation operator. Okay. But your theory is not time translation violence. So what it does is because if it were translation in violence, it will be identicality. So it's telling you that including the boundary. You can take anything like, you know, you appreciate the boundary as well. No, it's not it's not telling you it's just telling you that the really important contribution come from really the boundary so that's the thing that because at the end of day, what this is telling you is that because you see in this computation you, you get your original function you want to compute as something for which the only contribution from the bulk to bulk propagator that you get is from the boundary which actually allow you to split it so in a sense it's what is telling you that the information of a big graph is encoded by putting some of the internal by putting internal at the bulk to bulk propagator to boundary. That's what's telling me. I'm not telling you something about translating the full graph by itself. No, the fact that you choose the operator in a sense it's a trick because besides you know that there is two out of the terms of the bulk to bulk propagator that are going to be cancelled. Yeah, exactly. You can make this trick with any integral with a G which disappeared at the bottom. Yeah. I think the microphone is good. The microphone is good, and I am about to switch off. This is an option 23. Okay, welcome back to the second session. So as you know, unfortunately, we had a change in schedule because we have some type of rooms, but we're very lucky to have a speaker that could chip in and focus about exciting projects. And so we'll have. We'll have Julio tell us about positive geometry he'll give a very gentle introduction to geometry. And, and there's a lot of papers coming up so yeah very excited about that. Okay, thank you very much. So I will do my best to make up for for the missing of Neema. I don't know if I will succeed, but at the very least I can guarantee you that I will be done in 25 minutes. So, and it's not because I don't have enough things to tell you about, but it's just because I thought that maybe this is a topic that is not entirely familiar to most of you positive geometry. It's something that comes from the world of scattering amplitude, but since the seminar work of Neema and Paolo. And they've made an appearance also in the context of cosmology and even to understand that it's becoming more and more applied there. And so, as the title says, it's going to be a very gentle interaction, and Paolo informed me that he will tell more about his stuff in the coming days. Okay, so, suppose that we want to compute a scattering amplitude. Then we know what, what is that we have to do with some overall possible finance. But of course this is an unsatisfactory expression for the scattering amplitude, because there are many situations in which this country amplitude is a very simple object. It's a very simple function of the kinematics. While the representation of the sum of a fine and diagram is a monster. The reason for this unnecessary complexity. Well, there are many reasons, some which are more subtle than others. And sometimes the reason is that the scattering amplitude is simple because it's constrained by some interviews that you didn't know about things like you are conforming variance. And you didn't know about them because they were not manifest from the Lagrangian point of view, and because they're not manifest from the Lagrangian point of view, they're obscured from the tool that you get out of the Lagrangian, which is the representation of the fine and diagrams. So the amplitude is simple because it's constrained by the symmetry but the symmetry is destroyed by the fine and diagrams and so the expression on the right has to be complicated. There are less surprising reasons for which the same phenomenon happens, it could be that you're working with the gauge theory and then the amplitude has to be gauging variant that you know. But even so, each fine and diagram individually is not gauging values. So the same reasons that there must be a very complicated pattern of consolation between no gauging variant object that produce a gauging variant one. But I would say that there is an even more elementary reason for for what is, for what is from happens, and I have to say that elementary is a word that is going to be recurrent in this talk. And the reason is locality in unity. So fine and diagrams make manifest locality in unity, which I will make more precise. Let's consider a very simple theory. cubic planner scholar theory. The amplitude in this theory, as you know very well, can have a singularity only when the total momentum carried by a collection of consecutive particles goes on shell. So if we say, for instance, we have a p1 plus p2 plus p3 squared. And this is going to zero in some massless theory. So this is a singularity, which means that the amplitude looks is blowing up. So this is p1 plus p2 plus p3 squared. This is locality. The fact that we have this singularity. And the residue of this poll. It's a product of simpler amplitude, the amplitude factorized into, for instance, three level, three, sorry, for amplitude times another level for five amplitude. So this optimization is what we usually call unitary. Now, they need to make manifest both locality in unitary. Required for us to, to produce the sum of the time and diagrams because once we make manifest a simple poll, since we are factorized into other amplitude is up with the kind of other poles. So you can, after you have gone to this. In fact, you can take another collection of particles that send them on shell. So it means that this amplitude has to factorize as well, this amplitude has to factorize as well, and you can do it in many possible ways. The only way to make manifest all these possible patterns of repeated factorization is by some of the final guidance, which is what you get when you factorize as much as you can. Okay, even if you start with a reasonably small number of possible factorization channel something like n squared for a planar amplitude. You end up with a very complicated with a very large number, roughly four to the n number of final diagrams. This also suggests that maybe you try to find the representation of the amplitude. You may try to write the amplitude as a sum over the factorization channels. And you may wonder whether this gives the right result. Now again, in complicated theories there are many reasons for which these are simple formula fails, especially if you're spinning particles, but even in this very simple planar cubic scalar theory. You can see that something goes wrong. And it's just that you over count the possible time and diet so there will be a final diagram that appears in the channel and also appears in an hour. So this is again the wrong result. But, okay, this just means that this simple name approach doesn't work it doesn't mean that maybe you're not able to find a formula like this one, but you have to think car. And let me oversimplify a little bit, but by thinking very hard about this question, especially many for for your meals, many representation of scattering amplitude that we found. For instance the BCFW occurred through the relation, but then you end up you find other problems you have to introduce spurious polls you have to understand the cancellation of the spurious polls. And long story short, the organizational principle that tells you what is going on. It's a geometrical object in any word for was first discovered in any word for by Nima and the yard and was called the amplitude. It was discovered first in that theory but then we have we have seen a never growing a list of examples where the same idea applies in the same description of works. And changing some details. So let me give you a very brutal. This simplification of this idea. The idea is that for every amplitude in your theory. There is a companion geometrical object positive geometry. There is some geometry defined by the positivity of certain function. So maybe you ask some linear function to be positive or some more complicated linear function to be positive and the region where they are positive positive defined some object. This correspondence makes so that every amplitude is mapped to a geometry and every possible factorization channel of that amplitude corresponds to one of the boundaries of your geometry. So for every singularity there is a boundary. For every singularity the amplitude factorize and the correspondingly the boundary of your positive geometry factorizes into the positive geometries, which are associated to the same to the factors of your amplitude. Okay. So here, turning these nice words into an actual prescription to repeat an amplitude. Let me show you what I actually mean. So I want to show you an example of this correspondence between geometry and amplitudes. Again, it is a simple. The theory that we will consider is we call it the trace of IQ theory, but today we're going to talk just about the pre level. So you can think about it as by a joint theory is the same theory is by a joint theory. But if you want to complete a main point amplitude you have to consider a disc with 10 market points and draw all the possible five and I am which have external legs on the boundaries of the disk. So something like this. Well, it has to be cubic. Could have done something similar I guess. Okay, so this is one diagram and you sum over all of them. Yeah, I don't know why I've done such a complicated example. I'm more manageable one. Let's say five points. What I wanted to show you is that is that every diagram is dual to a triangulation of the disk. So this is the diagram, and this is the corresponding triangulation. And double the market points 1345. Every propagator is going to be dual to accord that connects to the market points. Very good. So, how do the amplitude in this theory look like the simplest one the three point amplitude what there is a single diagram in a sense. So the amplitude is just going to be, let's say one. So, at four points, we now have two diagrams. And, well, usually we say that they contribute with the SMT. It's going to be convenient for us to use the same labning of the propagators as given by the dual angle. So I'm going to say that this is a X one three. And this contributes with one over X one three, and this contributes with one over X two four. So these are the two diagrams, which are dual to either this triangulation or to that. Okay, and what I mean by X one three is the propagator that is dual to this angle, sorry to this cord. And in general it's going to be given in terms of the external momentum by PI plus PI plus one plus dot dot dot until J minus one squared. Okay, so this was the four point, the five point that was supposed to do with here. Well, we have again that diagram over there. We're going to give us one over X one three times one over X one four. And then you have a five other diagrams, which are obtained by secretly rotating the triangulation secret. Okay. So what are the geometries that goes together with them. At three points, we have nothing for the amplitude and accordingly we will have nothing for the jump. It's just a point. For pointer, we have to have two singularities X one three index to four, we call that singularities are boundaries. So we have to have two boundaries. Simplest thing possible is to have a segment. And at five points. Now we want to be able to reach two boundaries so we can set X one three to zero and then we can set X one four to zero. This is something that has to be two dimensional. And that's to have a total of five boundaries. So that's a panda. This could be X one three extra four extra four extra five and extra five. Okay, and I'm going to draw for you the six dimensional example, the three dimensional example, this is going to be the highlight of the talk so this. Yes. It works. Okay. And now let's look at this guy for a second. This is the amplitude at six points. We see that we have a bunch of pentagons, why we have a pentagon, because the six point amplitude can factorize into a three point amplitude, which is a point, which is a point times a five point amplitude. This is a seven particle, and the five point amplitude was given by this pentagon. So we have to find among the boundaries pentagons. But then we also have squares, because of six point amplitude can also factorize into four point times a four point time. And the four point was the segment segment times a segment is a square. I'll leave it to you to understand why we have six of them. So this went a little bit faster and getting better and better and drawing it so maybe it's easy to miss the point. And it's that it's very, very remarkable that you can actually form a polytope out of the possible single items. If you think about it you could, you could have started from a single art and put a pentagon there and try to put everything else, but there was no guarantee that in the end the picture would close and form a polytope and not something with holes. Okay, but he's a very nice pictures, but now we have to understand how to get some physical quantity out of it. And the connection between a geometry and the natural amplitude comes through what is called a canonical form of the positive geography. Even a positive geometry. There is an associated differential form that lives in whatever space the geometry is in. And which is uniquely defined and fixed by, by requiring it to have a singularity only at the boundary for every boundary of the positive geometry, it has to have a simple pole. Like the boundary happens at, let's say y equals zero, the differential form has to look like this. Omega has to look like that if y equals zero belongs to the boundary of the positive geometry. And furthermore, the residue at every boundary component of your positive geometry. So we see that it has a simple pole, but then there's something else that something else is the residue. And it has to be the canonical form that you attach to the boundary itself. So this is a recursive definition that starts from some high dimension goes down dimension until you find the code, maximum code dimension zero logic points. So you require that the canonical form. What's here there is a plus or minus. You require the canonical form of a single point to be either plus or minus one. Okay. So, in general, it's very difficult to compute a canonical form. So let's look at some simple examples for the simplest geometries, which are politics. So for the positive geometry is defined by the positivity of some linear function, all the points why such that some linear function is non negative there. So the simplest, the simplest example again would be a segment. So let's say that we have a segment with vertices at a and B. What is the canonical form. Well, it has to have a pole of a and it has to have a pole of B. So it could be something like this. But then recall that we want to raise you up points to be either plus or minus one so you need that normalization factor there. And now it satisfies the requirements. Okay, this looks very simple so we could imagine that maybe the simplest is the next to simplest example which will be something to the national something like the Pentagon world in the court. So it has to be as simple as let's say that he lives in some space. Projective variables. Why this is a canonical formula is a canonical measure that you have to project a space if you don't know what it is it's not super important. Let's say that the boundaries corresponds to some. I per plane, even my linear equation, then you can you could wonder whether this is the right result. After all, we want to have it five poles at the five possible boundaries. But now this doesn't work you cannot engineer a simple numerical factor that doesn't depend on the wise, because you see this form as a pole here and it has a full there. So in this point here, where the planes meet the canonical form as a pole. Sorry, if I said that that cannot depend on the wise I said something wrong. I meant that you cannot simply put a constant here, like, in this case, the numerator didn't depend on X. Yeah, you have to have something that actually depends on why cannot be just cannot depend on the dappies. So the root of the problems we were seeing are these five points, the form as a pole there and we shouldn't have a pole there because those five points are not on the boundary of the positive geometry, but you can do something you can take the unique order it goes there. And you can put it in the numerator. So this puts a zero there, and it can set the pole from the denominator. So this is okay. Okay, this was just to show you that you see many things look very simple when you write them down but when you try to do example you see that these are actually very intricate object to compute. So this is a systematic way to compute the canonical form of any positive geometry, the, the, the most important method is that whenever you have a positive geometry. If you decompose it into other positive geometries is doesn't really require the positive geometry to be linear. So the canonical form of the positive geometry is the sum of the canonical form of the canonical forms of the simpler pieces. Very good so this is a formula that we will see in action in a second. Okay. Now, let's recap a little bit what we said so far. Yes, yes. This one. Okay, this is just we're working in projective in homogeneous coordinates. So, this is in P two. So you have homogeneous coordinate y zero y one way to this measure is defined if you want. Okay, homogeneous coordinate. And then you have a fine coordinates x i being y one, where y zero, and that's it. And this form is just the pullback of DX one DX two. And that is, and that is. You substitute the access the ratio of the wise there. It's called the oiler form is a standard measure that you have one project space. It's important because if you want to build a differential form, which is well defined on a project space, you just multiply it with a function which is of the great. Well, sorry, of an homogenous with an homogeneous function which such that the overall weight of this is zero. So for instance, here we have a, if you rescale why by lambda you get lambda cube lambda square is lambda to the fifth and you get five, why is it so you get lambda to the sixth. So this is what I. Yes. It seems to me like that example the only possible in beauty. In this example, yes, yeah, there is always an overall minus size. Well, if you, there are certain technical requirements, you, I can tell you more about it later but yes, it's the only thing. So just to recap a little bit, we have seen, we have a theory where at least in a few example we have seen that there are corresponding polytops meaning that they have the same combinatorics as the singularities of that theory. We know how to. Well, we know that the polytops is associated canonical form. And now we will see how the canonical form of certain polytops, but those that I was doing before, give you the output of that theory. But first step now I have to tell you how to actually get those polytops and not just the pictures. So, and how much time do I have left. So, because of the simplest polytops was just a segment, which was supposed to have a boundary of x one corresponding to x 13 and the boundary corresponding to x 24. So how do you get a segment with two boundaries, you can simply look at the positive region where one of these functions is positive. Sorry, where all of these functions are positive and intersected with a plane, which guarantees that they cannot go to zero single time. So if x 13 zero, it cannot be that also x 24 is zero. So why do we get a segment, why do we get that segment because the positive region looks like this. And now we intersected with a plane. So we get this segment here. Okay, this is the basic idea that creates all of these polytops. I should mention that these are called AB, this is the ABHY realization of the associated, which is the name of this polytops. Okay, so in general, we are going to build a nice triangle where we record all the possible chord of the angle recorded a chord of an angle and correspond to the singularities of of an amplitude in this theory. So, for instance, we could have x 13 and x 14. This is mesh. Well, every time you move in this direction you increase the second index every time you move in that direction you increase the first index so here you would have x 23. If you recall the picture that I was doing before x 23 is a is a boundary chord of the angle so it doesn't really correspond to a propagator it's the word for external leg. So the quantity is supposed to be zero. So extra three x 24 extra five. And x 35 here we would have x 155 which is also zero and x 34 which is also zero. So at the boundaries here we only have zero and the five possible propagator lives here. So for every mesh for every little square that you have in this picture you impose an equation like this one, which keep a keep apart to incompatible boundaries. Whenever you have something like this. If you write x i j plus x i plus one j plus one minus left minus right is equal to a constant. So for instance in this case we would have x 14 plus x 25 minus x 24. We would have minus x 15 but this happened to be easier. So you, you ask this to be some positive constant let's call it to see one for using the same number as the bottom corner. We asked this positive number this constant to be a positive number. So this means that the variable x 14 and next to five cannot go to zero simultaneously. If x 24 has to be positive as well. And if we go back to the picture of the angle 14 and 2512345 correspond to crossing course. The crossing course, obviously cannot be do well to propagators that you're on us on some diet because that's our five man diagram is the world for triangulation and a triangulation is built out of no crossing course. So we don't want these boundaries to meet these are incompatible singularities. And this is a question guarantees that this doesn't happen. Okay. So this is going to be very hand-waved going to prove that is actually gives a realization of the political that was done before but this just gives you the idea. Now finally, we're going to compute the first amplitude of this talk. We're going to compete the four point amplitude. So attached to this political here. We could solve this constraint by saying that x 24 is equal to C minus x 13. So I'm going to withdraw the political as having a boundary at x 13 equal zero. So this is the space x 13. And another boundary at x 13 equal C, which corresponds to on this point x 24, which is equal to C minus x 13 is zero. So before we were down the formula for the canonical form of such a segment. It was, it was me minus a so in this case just see over x 13. And then we have to put the other boundary so C minus x 13. It's a differential forms with multiplied by the x 13. And so this is a, this is the canonical form. And the claim is that if you take the canonical forms, and you divide by the measure. So you strip away that factor, the function that you're left with is the scattering amplitude. And he did. It is because this is just the same as one of x 13 plus one of x 24. And that x 24 is given by that. Okay, so a lot of work to get the usual representation for the amplitude, but now let's look at the Jewish example the five point. So, and we're going to find a new formula that doesn't look like a sum of the final diamonds. So if we write down the equation corresponding to these guys. That's very quick x 13 plus x 24 minus x 14 people see one three greater than zero and x 24 plus x 35 minus x 25 equal to C 24. So we can solve this equation and express all the axis but x 13 and x 14 as a function of x 13 and x 14. And you plot the region where the corresponding function are positive, and you find the following policy. So, Okay, so this leaves in the x 13 extra for space. This is the boundary where x 13 zero, this is the boundary where x 14 zero. You find that this is the boundary where x 24 is zero x 25 is zero and x 35 is zero. So you see that, for instance, the boundary x 13 and x 24, they do not meet on the political. You have to actually do it to see it but it's, it's a very simple thing to do. Now let's compute the canonical form of this object by doing some decomposition. So for instance, the particular shape of the politics suggest to take this the composition. So we divide the political in P1 and P2. And the reason why we do that is because each of these pieces look very simple. It's what is called a prism meaning that is a, it has a bottom facet and an upper facet which have the same shape. So you can imagine that the canonical form of this guy is going to be the product of the canonical form of one facet times something that puts a poll at the bottom and the upper facet. So for instance, for P2 five, five point. Sorry, omega of P2 is going to have a poll at x 13 equals zero up all x 25 equals zero. And then here we need the canonical form of the of the facet x 25 equals zero, but because everything is completely recursive. So the canonical form of this facet is a political associated four point amplitude. We have computed the four point canonical form here. So, and the canonical form of the of this facet is going to be essentially these after you can label the variables. So, this is going to be one over x 35 plus here there is a little surprise because this was a form living in a one dimensional space. Now you think of this form as living on a subspace of a two dimensional space and when you do the pullback to promote it to form on a full dimensional space, you find a slight deformation of that formula. You find instead of having x 24, which would put to the pole just here, you find x 24 minus x 25 which puts a pole and all these internal battery. So you find this form for the other piece omega P one, you find the extra three plus x 24, one over x 14 plus x 25 minus x 24. Now, the amplitude is the sum of these two pieces. So these are functions, so maybe I call them like this. And now this doesn't look like a sum of a fine and diagram for a very simple reason that you have introduced as poor as poor at x 24 equal x 25. And of course if you sum over final and diagrams you never introduced by definition almost as poor as poor. So let's find the four times from one side from another sign, and there is a cancellation that makes sure that the formula collapse to the usual five point representation. Very good. So, how much do I have left. I don't have any time left. Good because I'm not. Well, no, I wasn't done but I am done. Maybe I will just tell you something real quick about the conclusion. Here it is, was the last page. I just wanted to tell you that all the thing I said so far is in this simple case of a three level planner amplitude, but actually generalizes to every order of the one over an expansion in this theory. So it doesn't rely on the planner on computing just the planner limit as in the usual and equal for super young years case. And also, you find recursive formula for interns, but you can also compute the loopy internal and find interesting formula for a national amplitude. Thank you very much to Leo. That is really great. Thank you. Yes, so there are many versions. I think a pinnacle energy was one of the first to do something about it. And there is actually a version that Neymar was telling me about Neymar was telling me about many things as a shame that it's not here, which actually is able to write down any fight to the P interaction even mixed interaction, using this story. So from the, from the poly top of the cube theory. And then there is also, but that was another question. Yes. So to be more precise what he produced for you is a. So you know that for a single fine money there is a final parameterization. Right. So there is no loop in that expression of course there is the part of the parametric integral to be done. So this for you is a unique integral over a space. In a sense the space is the dual of this policy of an interned, which is a space was linear on this space, and is not defined by something over the final dimes. So if you want to evaluate these intern on the point of the integration space, you don't have to write on your computer, a code that generally sold the final diagrams and computes a value for that particular point. It does so in a way that scales with the number of single artists of the answer, other than with the number of final dimes, but then you do this in turn you can do it in many ways you can do it in Monte Carlo, for instance, and you get an output, literally a number which is the What did we have to say that you get another type of type of fine man parameterization however unusual one and one that's valid for the full amplitude. Indeed, yes, the best way to say is, as the answer said it's it's a global string of parameterization of the amplitude. Sorry. Actually indeed I mentioned now the thing is that, of course in general will be divergences so you can compute globally parameterized dimension regularly is a regularized amplitude, or you can do it in two dimension where actually, there are no divergences if you exclude the surfaces. I didn't say the word surface before but for every surface there is an accompanying political. If you exclude surfaces which have functions in the bulk, if you only allow functions on the boundary. So the duality is gives you diagram without that. This gives you that also if you don't use these ones. If you ask to only have market points on the boundary. The dual diagrams are convergent into dimension so you can compute that dimension and it gives you a number. Yes. All plus. But this is a this is a scholar. There is no electricity here. This is a literally five cube theory. But I think the claim is that you used a d dimensional integrants. So for the remains a question of renormalization and so on but it's, it's not only an individual dimensional. Yes. And I mean, we're not completely crazy. The reason why we care about this theory of trace of IQ theory is because it's the theory that has to do some overall diagrams that you can draw on a surface. Young meals in a sense is also theory like that the difference between young meals and trace of IQ theory is that you have to add interesting numerators. How to treat the dos numerators in this language that we don't understand yet. But we're optimistic. Okay. I was also very curious about this comment. I think it's really, I want to thank you, especially Julia that you, you agreed to give this talk on their short notice. Thank you. Thank you. Likewise, what I just said, that's all for the next speaker who even more than, than being requested on short notice was even given some, some wishes for what, what a possible topic could be. So, do I assume correctly that you will tell us something about infrared diversions in the context of gathering amplitudes and cosmology. It's wonderful. Thank you very much. So, please take it away. Thank you very much. So as an advertiser, so I'm going to talk about infrared division and actually what I'm trying to tell you some story, which mostly will take place on the other side. But actually, during the lecture, I learned a lot of new things about cosmology side and I believe I hope the lessons which we learned about gauge series will be helpful for cosmology as well. And you should forgive me if I will tell you something about cosmological cosmology side. But once again, I'm just a process of learning and sense of excellent power lectures and some nice seminars they've had during the week. I think I understood something. I just started kind of summarizing what we have heard this week. So I'm going to talk about cosmological correlators and people have seen already bunch of diagrams appearing for example the diagram which appeared in lecture. So you're talking about production of particles out of particular vacuum. The case of the city that was a bunch of days with you, but in general, as from what I understood, it was familiar problem in general terms that there's the cosmology, it has some kind of background. And you look for the production of particles out of this kind of the ground, which is detected certain time, and here you have particles, we should read three and four, and then I saw this object goes under the name of cosmological correlators. And then we can see in power election, and also guess that you need to talk. If you look for the properties of these correlators as a function of the energies with particle produced it has certain political properties. One particular lesson, which I learned. So if you look particularly as an example of the total energy of the system. In general, it will behave as a total energy in some power, and what appears to be an emulator, come out to be on cell for particle embryo. So in this case it will be for particle energy to look like that. And then this way, to me, this is a relation which kind of connect to different worlds, the world of cosmological correlators, and the world with static amplitude. And yesterday, during the cost of he actually mentioned that there is kind of one to one corresponding to the observable to define on both sides cosmology side and the case to the side. And then what's so called in out matrix element, which we're exactly scattering amplitude. And then there was so called in adorable. And the cosmology sides, which were corresponding to total process. So it would be kind of to tell you what happens on the left hand side or on the right hand side of this component. And hopefully by the end of my talk, I would like to go back and tell you something about what happens here, if you start going beyond three level of estimation. But I mean by three levels of summation, I really want to talk about those cosmological correlators by going deeply into level and trying to see what kind of structure you're expecting to find the level of the loop. And I wouldn't answer that question but I will answer the question what will happen to start looking for properties, namely with separate here. And hopefully by the end of my seminar you will see the answer with going here it's very simple. So but let me start slowly. And as I told you, I will start from the other side, and the natural things to consider which was analogy with the process should go here. It's a process which goes under the name of the E plus minus any donation in the head. Let's talk about this particular process. And the reason. The process is actually a tool for one hand side one of the most best time for us is to see if we know a lot of things about the process. And second reason is that if you actually look how this process looks like, you will see there is very direct analogy with what happens here and what's going to happen over here. You start from flat space, everything is stable, nothing happens, no particle production, and now we want kind of to create something here. But instead of putting the curve space, I excite my vacuum by certain operator, if you wish by producing some virtual photo. That's minus duration, visual photon in physical terms that correspond to station when you excite excite vacuum by applying electromagnetic current, the certain energy or mental cool. This energy gets released, and it released to produce two final states. And in this particular picture the final states will be pair work and under. producing a lot of radiation, for example, each of them put in the glow on. And in this case, what you're going to have for this particular diagram, you have production out of the vacuum. And those four particles is actually similar to particles. In this case, you have one to one analogy between production of particle in the physical space and production of particle in the flat space out of the way. And because of this relation, you're expecting that there will be some kind of cross talk between the different. So now let's go deeper into this object and try to see how you could actually compute this object. There's no way to compute this total cross section. One way goes through this formula is in our formulas, or another way of saying you could decompose total cross section into some of the final states, let's call at all possible And in this way, the total cross section will be some of the total final states and each final stage will contribute to the matrix element of the condition of the initial state, which is important for the refinement. And then this particular case, if you just start thinking about the value of the expansion, so this becomes a formal diagram, which will put on the work. Then you will get the diagram, which corresponds to the initial one glow, and so on and so forth. So then you take model square because the phase space and you will get what you will get. And let me mention that another diagram, which will be important to follow, which correspond to the results. This is our kind of classical textbook. So this was way number one. Which goes to the amplitude, but there is much more efficient way, which also was mentioned yesterday, which goes on the name is optical series. And what stadium tells us that the total cross section is equal to the major part of a forward static and, in other words, set of going through the amplitude let's be considered as a particular process when you have visual photon goes into everything and goes back to itself. And then you compute this object. And this object I mean, time for the correlation function of two colors. We're going to add zero. You take this relation function, you validate it deeply in your creative space. There's no diversions is everything is fine. You take four years of school. Then you don't look at the nation from negative square to positive square. You take it to major part. And this object by optical series will be exactly the same to cross section. So if you do calculation second way, you will get perfectly fine at number and actually this total cross section. So sigma not correspond to the bone for sections of the last half here. Then you include one more here you do the calculation you find the payment result. The one cross section is proportion to alpha s strong, strong coupling constant divide by five. And it will continue so far and so. And I believe by now, you know, analytical expression for the procession up to follows. So this is just another way to tell you that this second approach is quite efficient to go loop by loop doing calculation without uncounting any problems and any stable calculations for dimensions. There is no even point to talk about any regularization whatsoever just perfectly fine calculation which we're doing with CD, or the back to the correlation function. But now let's go to the first approach and try to see how could reconstruct the same number can go through the epic. And that moment you need to run the trouble. As a trouble arise, you can print the image. So how do we see this divergence appear. Let me start from the last diagram, let me turn it a little bit like that. Again, we're talking about diagram it has two particle produce to coax, and then you have some virtual glow in it. And obviously contribution of this dial proportional to the power of the constant is integral of the face page. And then let me be a little dramatic. I will put you. I'm going to keep only one with 10 which will be relevant for discussion. So these are approximately for. And then as you know very well. What's happened next, because now I'm going to take into account that you want to be to own its muscle, we take off muscles. You could simplify the expression a little bit. Now you see the problem here. So the problem is that if you start computing the cynical immediately realize that it's still the ill defined. The problem here from two potential dangers region. The first is the region when he is small, so we'll moment the case more all components are small. This thing is where obviously will be so we can compare the living term and then you will see for powerful. For powerful mirrors is a signal that you have the vision, since the versions come from small kids is definitely infrared. So there's a need that will give you a call. It's a parameter of a solution, but this is not the story because if you look once again at this integral, you'll see that there is yet another agent will come if he is not small, but if it's like this to one, since to one is like like factory need to see the light one. When it's when it's long. This addition to produce the vision. Similarly, if he is calling it to you will get yet another contribution and totally double fall. What does it mean that means it just starts from very simple diagram at lowest of the potential expansion and you really see that at that level. And then you get that user mission organization. And then the day you get the virgins, which is double one for because of infrared, the second because of the calendar. So this is a synonymous of saying that as soon as you have massless particles propagated in your series, you should expect him to get into troubles and these are troubles. But now if you come back to the first slide, you need to run this problem because we got two diagrams at one level. You know the sum of them have to be finite because I already gave you the answer. But that one is the virgin. The vision as well. And it is the vision because you have to take the division of this diagram is limited to one. What you have to do you have to repeat the same population but this time, you have to take this diagram. Take it more than one square. This is your billing. Okay. This will fly to modify your integrant. Come now. What is where to find class request is a fact that the balloon in our going against energy is positive. As they could perform similar analysis to realize and using the process also the vision. One of it won't work with some sign. And if you could guess if you if you now add together this division division that then you'll get something finite and then we could construct the final expression, which will make exactly the one which was coming from the calculation. So this way from this simple exercise what you see that if you do calculations to the amplitude it will start dealing with this. It will introduce the national organization intermediate steps, but the final result is better to find it matches the result of the calculation of the relation function which was done entirely for them. That's another way of saying that if you always be like the so called important qualities which total intersections, they're much much better suited to doing calculations because basically you don't have to deal with those different divisions which I describe you. I just want to understand what the physical high those divisions, why do they appear with the general rule, the visitation you expect them to appear. And the answer to this question has been given a long time ago by Lee now recognize it. I will talk about this year but a long time ago. And what they found they found quite interesting things, namely, and what they're saying that he was thinking about the diagram, and we agreed that the version of competition when case soft or case collinear to IZP one and two. In a physical term, this corresponds to a very special situation when you find a state becomes degenerate, what I mean by this. So imagine you want to define this object as mathematics element between virtual photon states, and in this case of the state of the cork and the cork and the goal. If you're thinking about state of the cork you will specify the state by giving its momentum and this quantum numbers. But now what will happen if I will secretly add to this cork some balloons with zero energy, this energy zero this particles and observable. But if you're thinking about cork and part was going to zero energy for point of view of quantum numbers in the same state. So therefore, strictly speaking, if you go to the limit when case soft particle assault. You have degeneracy between single particle states. And work with the global medical entity. In the same manner if you now go to the colonial limit is it blown is aligned with the core. You have exactly situation with the two particles which align. You can distinguish by any experiment one particle from two particles which have the same. So this is actually situation we have degeneracy of the final state. So this gentleman is to say that every time you have degeneracy at the level of physical states. You bound to enter into this drama. And another way to see it's you don't need to go to one of your series actually this phenomenon is known in quantum mechanics if you were doing to do but expression quantum mechanics this problem is known as a problem of smoke. Let me give you some flavor how it appears. If you have some kind of mechanical system. We describe a certain kind of tournament and you want to find ideas where you can I can see it or listen. You could think about this and Antonio, like describing this asymptotic states in your theory. Please. Generate state. If you have a quote, what has talent charge, especially like a charge. But now if you have to walk. You name it energy momentum electric charge, etc, etc. What I'm saying is if you're now adding some blow, which goes say momentum is it momentum to light or that more. There is no way you could distinguish these things from that one, if they have the same level charge. Oh, I thought you were saying that the initial state that it was a different. What I'm saying is that if you're thinking about this amplitude have initial particles momentum to one. And another let's say you want minus alpha k, you use alpha k. So you have initial stage which was for single particle state for the moment that you want me just like like, and I'm looking for the final stage which now consists of the folks and go on. This is the momentum of the bp one minus alpha k and going to alpha k. And now what I'm saying is that this k, I could choose to be aligned along the one direction. So I have two particles in the same direction. Then the states and coming to the going states have exactly same quantum numbers. You're saying that you should add up to me. These are what will come of the recipe at the end. But the reason why the region is here. The line reason why did you just sit in this because there's the generous of these days together with that one. For example, you mentioned if for example if glue had a mass, you go ahead and mass. There was no way you could have particles. Masses particles split into masses, but it doesn't exist. It's only part of the master. It's because of their masters you generate divisions, but once again the reason, the starting reason is exactly because of the generous and initial final. And just to give you a solution for this actually you kind of already seems to some kind of. Again. Okay, okay, okay, okay. Let's see. So imagine you have ingenuity. And to empathize. So, and to empire. And I'm saying what will happen if I have here one additional part of the final state. And now, if you go to station we have here it correspond to the station when you have the same initial state but this final stage with the physical side degenerate. There's only me. Which region. If you're talking about local region. I think it correspond to the exchange is any is called initial state interaction when you have a change on the glow. And this glue will have most of this first moment. But it's just a matter much smaller than a matter of time. Therefore, in the limit when energy of the particles goes to infinity, it's a station for the power. So all the reasons is if ready, which is the linear division to go out by the vision is exactly the same again. And let me just finish with fun we can exercise is probably hopefully it will be very clean. That the major you have some kind of mechanical system. And you want to look for solution to the system and you know that the look at the commission power. So then we're going to and the physical terms. This is going to be the spectrum of particles. And now we're asking the question what will happen if I turn on the direction. How does the function will change how does it change. So when you look for the business TV solution, you will find the image of the state. To get the correction, which is expectation where they're on the probation. But if you look for the changes in the function, it will be more complicated. It will be infinite so all states. So you start from one particular states. You ask the question will be transition amplitude from the new states to original one. So there's a new stage will be given by the sum of all states. And now you look at that expression you need to see the denominator and the denominator tells you how your states which are talking about different compared to other states. And this is exactly what I was saying that if it happens station that this denominator is very small, you're in trouble. And the situation we're doing here is exactly the situation when you have this energy and energy, which are differed by additional ones of particles. These are all trouble stuff. So, once again, the vagus is collinear prejudices the synonymous of having these degenerates in the final states and the ones again this gentleman and now in a sheet. They also gave you prescription. And now you could get rid from this problem small denominator, or another way how you could get some physical, final physical observable using the agent and because you know very well. There's a very different prescription which goes on the name of summation of the final initial state includes enough if you just summit all final states. If you have on this station, which will put on going on the number of particles, if it's efficient to include your state of final state. This here you will tell you the fine result will be fighting. And indeed it's fine as I told you, and then there you will reproduce total cross section which comes from the calculation. Okay, so how do you start from this point. You see no interaction, three particles. You're defining your human space will consist of one particle two particles in number of particles, each particle coming with its own momentum. Then you're saying what will be the transition everything like picture which I raised before. I have traditional like that. And I look for my initial states which are called here, so I am going to some final stage which are both. And inside this book, you have the interaction be here over here. So then I, if you apply that formula, you will have to some of all possible state, which appeared with this formula. And the native will come with a different between initial finals. Now what I'm saying that if you now have particular final states, you can choose the final state now you're creating some additional more particle, additional particle would contribute to the sun is yet another contribution with one particle more. And then it will also contribute over here. And what I'm saying is that now if you have a situation where n, the states and states k differ by some small amount, the small amount will be due to the mission of this software. And that is zero and denominator, and zero which created divisions. I would have a senior. I see that there is a very good thing that if I send over the final state to the urgent. Yes, yes, if there's some conclusion of what is there, the zero that counts. Yeah. So this gentleman was saying, so he will start from this object here. And now you want to look for the total cross section. So this, then it will be as far as projects. This, this sum in particular states and addition matrix element between initial state and states game. So forget about the sun. To get total cross section you have to take a model square of this object and some of all final states. In this case, you will have double some amplitude. Take it's models for a new song. Then what they're saying that if you will some of all final states for the key. And also, over all initial states for all. Simple manipulation with some will tell you that the day you will get zero. One of the end of minus k plus one of the k minus. Sometimes like that. Sometimes like this. There's a more or less what will happen if you're some over both initial in the final state. I come to this, I come to this. But the reason the initial serum serum you turn in our breaking the sheet requires you to perform summation for both initial and final states. Okay, but then exactly what you're saying that in my example example gave you before I started from the initial states with there was no particles. It was kind of initial states was kind of coming from external virtual photo. And that basically optical Syrian told you that you find this out to be fine. These are what called blog Nordic theorem, which goes atop awfully now we're saying that if you're dealing with the cross section initial state doesn't contain particles, which could potentially produce the differences. There is no need to some of the initial state which is efficient to some of the final. But if you have at least one part of the initial state which could potentially need those softener particles. No, no, no, no, then you, you run into problem with divergence. It's just the whole tree level. That's the urgent. There was something like the pencil that I run into. It's again. Yes. And you see that we have time to forward. So let's be more precise. Let's talk about that. Yeah, but I will take slightly more different process I think deeply virtual. I want to get a virtual quote on my quote. I look for this problem. This process goes on the name of deeply nice. People like to schedule you can compute for example the diagram for here, you will find that it contains both infrared and the linear division. However, if you will sum up this diagram is all other examples of what you can put your mission correction will find that that will hold and go away, but see before we remain. So therefore, in this situation, once you have initial space, which could be so far as there is no co oscillation for the force, there will still have division. Again. So by the Because with the vision. Each one is set urgent. But if you solve all of them, no, no, it's not true. This example, these are forward scattering so I have initial final state being the same. So you take a major part of the content of the section. And the process of the case when the focus matters is infinity. A simple form and the simple form. But then with the question. No more questions. My statement is just that if you want to answer this. So for example, yes, the only thing that cancels it is Ford said in that it wasn't E minus one to E minus one with the box that just like that. This is what people have been doing for the last 40 years. Because Let's let me first say what people know because he didn't then probably will come to what gets in the city is this part becomes part of the yet another formula, which called comes in the name of the picture. So the diagram tells you it tells that you have work, which is mostly for which case at the moment. This diagram will be the part of the full physical perception. The physical perception that's going to be corresponding to the time for them. The final contribution becomes integral for the issue of the moment, okay, where this work it's up to come. Scanning variable called T times the diagram which we're talking about which goes under the name of the four distribution inside. Which depends on the new because you was exactly this couple of reasons. Times this process which we're computing. So they see it's exactly goes into this process over here. It also contains new square. This is the diagram which I told you, according to the standard of the world. It's simple. Yeah, yeah. It's fine. Yeah, of course. So it's making a statement that if you want to use tail in here to cancel their assets. And that's really the purpose really just being here. So if you want to figure out which diagrams pencil. Then we have to think basically relies on if you wanted the whole process. Yes, we need the forest that. Yeah, I have different style different understanding of you know what you should see if you want to cancel the versions of the one we can pick it here that which will do the job. And while when you have initial. Diagram will go like that. So you have to put certain here we're talking about initial state being more, which means both the final states, but now we want to cancel the diagram. So at the initial state load, which goes together with sport. As well as your square amplitude. It also becomes the final states and I was in the breeze of space for this diagram will cancel the. And this goes in the language I was saying seriously that now you have to include not only some of the final state but also some of the initial degenerate states. The general state will correspond to the blue on the board. But this is not part of our physical process. There's a reason why the diagrams then go into final expression, and we should stay and wait here you have to be the person. How much time do I have. Well, we have some great questions. However, we're probably not your main topics, so I think it would be nice to hear what you have planned. Yeah, yeah, let's me have five minutes. Okay. So now. I hope, kind of like, I get you something about all these problems that the vagus is very clean et cetera. But now the question appears. Let's come back to question which I experienced previously. Namely, what will we hear in Greek. What will appear here in greater. It will be certain amplitude. Let's say, for particle amplitudes. And I want to go to improve the corrections. Basically, I want to go look by look and see what will happen. I'm going to get one more level. You have those virtual contribution. The diagram like that. It's like the head of finance. But for the anonymity without all the issues that they can't do very well yet. Once again. And then the kind of physical stuff. Okay. But in that this is the last once again, I'm not cosmologist, right? But I'm telling you what I know from field series. And then this correspondence, first of all, in your allies and talking about doesn't stick in the response to the observable bodies. For the PCN rate, there will be a chemical combination of what you observe the physical region with respect to the energy. And only have to do such kind of combination you will come up with static. From that point of view, there is no contradiction with what you're saying in the object which I'm dealing with. But in general, if you're thinking about the source of the vision, this from point of view of embedded person. Exactly given in mind what I told you previously. So you have the outcome like that. When here is part of which is detected. They're affecting massless particles. It means that you have exactly the same situation which we have engaged series. What I mean by this is, for example, here. I'm close to observable time you're changing by some particles, which is soft. This particle could potentially create your problem. I don't know what kind of problem created because I never did the calculation myself. But it's what it's true. They can now translate the diagram. We'll be here. We'll stay here. Exactly because there's a change. We'll continue with the division. I think that that's what I'm saying. I think that's what I'm saying. I think that's what I'm saying. I think that's what I'm saying. I think that's what I'm saying. We'll be here. So there are some divergence, but the colleague, I really think that a physical reason you would see trust when we do the actual. So you get a sense. That's probably the translation I've said, in order to get the right thing we wanted to do at the fastest limit, but in and of itself. Here with the algorithm, I don't know if you might recall my original, you won't see it because you were really, in order to see it coming up there, then you're going to have the right to see it. You really need to be right. But this is exactly what you're doing here. Yeah, yeah, yeah. Sorry, sorry. We agreed that the picture which is still left-hand side which corresponds to real physical situation. It's not the one which will be here in the numerator. There is a continuation goes from here to here. I don't know, as I promised you, I don't know the answer for the diagram here, but I do have an answer of what happens here. And let me prioritize the answer. That is, you will see hopefully in the time amount of time, actually once you re-sum all the three divisions and you go to the limit when Katov goes to zero, like Epson goes to zero. Which is another way of saying that if you go to the situation when you put here elastic amplitude, and here we agreed that the picture which will be here will be amplitude with a number of steps, no less, you'll find another particle detected. For given number of the 30 particles, it's a limit when Katov goes to zero, the amplitude vanishes. All elastic amplitude vanishes. That's for the problem. No, no, for residue of the pole. For the amplitude, for the amplitude. For cross-section, everything gets fine. Let me just elaborate on this and then I think I will finish with you. So the statement which I am going to make is that coming back to the video, in general, it's statement falls for any important nationality in Syria. So if it was computed, for example, any amplitude is fixed number of 10 molaps and you choose kinematics such a way that it will look genuine. So let's say all these external momentum both separate in space. Or another way of saying, all Malvistam invariance are different from zero. So these elastic amplitudes will have an effect on the linear divisions. For example, if you're changing some sort of glons between some external X or if you need collinear glons connected to the blue X, so it has a different division inside. But if you take now the correct Katov to zero, this amplitude will go to zero. And this is the consequence of the interrogation. Let me show you how it works. Let's come back to this example which I gave you before. You put it with the double cross-section of any plus or minus simulation. And then you look for amplitudes of the neutral photon. For example, you have a possibility of having no real emission. Almost, then you have a possibility of having, let's say, one part of the final phase, and so on. So in principle, you have here infinite sum from zero to infinity. And each term of the sum corresponds to very special amplitudes of producing exactly an additional particles. And now let's look at the sum. So because it involves amplitude model of square, you have infinite number of terms, each of which is positive definite. You have no question about it. It comes with all the particles here. So now let's look at them by term. So for that term, you have neutral emission, and then we have simplification. There is predivision, so here. So this object here, itself, involves some pulse of the parameter dimension of regularization. So now what will happen if I send epsilon to zero? So there is two options, which is zero. One option, it will go to zero. Another option, it will go to infinity. And third option, it becomes constant. Let's examine this option one by another. So if this term goes to infinity, you're in trouble because you know the total physical cross-section is finite. But here you have already infinite number of terms and you already identify the first term, which is positive definite, blows up. It's not possible. So let's examine option number one. So what will happen if it vanishes? If it vanishes, then it's perfectly fine because you have infinite number of terms, each of them is positive definite, and this way perfectly well-consistence if you take infinite number of terms, each of them is vanishing. If you take zero times infinity, you could get whatever you like, including the final result, which advertises here. And this is exactly what happens in position. And the statement is this is exactly what happens in generic HCI. As soon as you have amplitude in expansion, which developers read the versions, and they blow up in certain way when it goes to zero, the utility will tell you that actually each individual term has to vanish, which is the state which I gave here. Whereas the amplitude have to vanish because of the new thing. So what are you saying? Reliance of the data becomes the cosmology of a processor which won't execute or it won't run because it can't work. It's the same as the point. So it doesn't matter which thing you're talking about. So you have some observable. Yes. Which could be explained to a number of particle produced. The first test, those particles should be mastered. Because if you have massive particles, there is not even an issue in that way. Then you look term by term. For example, have a virtual emission, long emission, etc. And you look whether those emission potentially could develop in further versions. If they do, you use the same stage. It may happen, they do not. And we know a lot of example, such series of even those three particle masters, those who don't have infrared divisions, or that are only three of them. If there is no divisions produced, you on this third scenario, which I mentioned previously, when amplitude is finite, and then there is no problem. You have infinite number of terms, each of them is finite, your subs will get finite result. But if you do encounter the divisions, this is the legislation which we will do. And so basically coming back to what I was telling, Relish the cosmology correlators and the data doctrine. As soon as object GPS here is ultra-lampage, which will lose mass of particles, which will potentially produce infrared divisions. But this argument you can miss yourself, that as soon as you resum all the corrections, the versions is kind of combined together, most likely they will initiate, but the sustainments is one longer series, to finally to actually to suppress them. So for the vision in the context of them, that you should suppress them, not encounter them, but suppress them. These are basically what I would like to tell. I'm not sure how useful it is for cosmological side, but at least as a lesson you could learn from my experience with the data spectrum. Thank you very much Aisha for this wonderful talk and already had some inspiring discussions. We have, yeah, we have time for questions related. Can you explain again how, why it was in the third option, which for this term are finite, but it is more than that. Yes. Yeah, this is what I said. So yeah, maybe I explained it wrongly. You can see also, if you may have a theory, which contains master's particles, but still there is no different division. You know, the presence of master's particle spectrum does not guarantee that you will have a vision. And then in ancient, you do have a vision. Another way you will start growing diagram and for example, let me show you examples that follow the sky. So you'll have five cubes theory. And for example, I take every two like that. And I'm changing this color back. If I add the integral, then more than 70, which I was showing before. This diagram for master's colors will produce your double pole. So therefore, I cannot guarantee, right? So once again, I could make a theorem statement, but for me, we don't have such kind of situation gauge theory because of how to report on the national life or for the national for how to report. But if this diagram has interdivision system, then in my opinion, you're more or less in the scenario which I described in the last video, namely that you will start to continue any more particle, for example, like that or like this, no possible waste, you will not have interdivision. So therefore your amplitude in this particular setup will blow up for the biode. And this will blow up for the biode that will remove the particle. Then exactly in one of those scenarios which I mentioned, which means that if you re-sum those divisions that they will bound together. So is it 40 or does it matter? If you wait for 40, there is no divisions. The three divisions will appear only in 40. Because there is some key view that you can watch that things are getting stretched if I could hope that they're in correct. Things are getting positive, it's connected. So the theory that actually we're going to mention of this. What are these two things from B that they clearly are going to be? You should be careful. It's true that once you go to this explaining universe you effectively reducing number of dimensions you're feeling, but it doesn't prevent you for having a third division appear because the third division, they come from the chain of the particles whose wavelengths will be arbitrary large like size of the universe. So even though, for example, you have some particle which I explained and which move apart with the list of light they could still exchange a few each other by changing some particles. Do we have more questions? So even the notion of mist as part of when you did learning in psychology you would say that the particle that oscillates you have mass on their bubble or what do you say about mist in the particle and maybe lower than the bubble or the level of mist as part of it? Yes, yes, yes. I think for all of these, I mean when we see that the stochastic effects of the mind is smaller than the bubble which is having that. So it could be that the problem plays all of the body purpose is having mist. No, but to add to this, kind of back to what the grammar that you might finish here that the biggest thing in the very computing is the logic correlates. There's no particles which are on shot per se. So when computing the diagram because you have finite propagation particles there is no notion of particles being concerned. But what this relation basically tells you that imagine you computed the diagram including also particles in interceptor. What it will tell you is zero which appears here. Basically it will tell you that because of the very supercell analytical properties who play over this kind of weather will be slightly different compared to those which you would expect in English. What I mean is that instead of having simple pool you might have some additional function like logarithms of this energy in some power which will appear in this object. Such that after you take this continue there won't be simple pool. It will be something which definitely won't have the key. So which is another way of saying that even though here you don't have per se predivision six but if particle propagates addition in a long time it's not a shell but it's closed on shot. It's almost on shot. And there's almost what will serve a further greater. So there's evolution time which you wish. Somehow it should propagate into formula to replace in front cutoff. So you're expecting it's not in front divisions per se but it will kind of echo those divisions where your evolution time will replace the in front cutoff. Thank you very much.